Body waves & seismic rays

Fabio ROMANELLI

Dept. Earth Sciences Università degli studi di Trieste

romanel@dst.units.it

SEISMOLOGY I

Laurea Magistralis in Physics of the Earth and of the Environment

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Seismology I - Rays

Heterogeneities

.. What happens if we have heterogeneities ?

Depending on the kind of reflection part or all of the signal is reflected or transmitted.

• What happens at a free surface?• Can a P wave be converted in an S wave or vice versa?• How big are the amplitudes of the reflected waves?

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Impedance

Any medium through which waves propagates will present an impedance to those waves.

If medium is lossless or possesses no dissipative mechanism ,the impedance is real and can be determined by the energy storing parameters, inertia and elasticity.

Presence of loss mechanism will introduce a complex term.

Impedance presented by a string to a traveling wave propagating on it is called transverse impedance.

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�

transverse impedance = transverse forcetransversevelocity

�

FT ≈ −T tanθ = −T ∂y∂x

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟ = T ω

vy vT ≈ ωy

Z = T

v= Tρ = ρv

acoustic impedance = pressure

sound flux= ρv

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Heterogeneous string

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Displacement continuity

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Force continuity

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R&T coefficients

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Kinetic energy

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Potential energy

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Total energy

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Boundary ConditionsWhat happens when the material parameters change at a

discontinuity interface? Continuity of displacement and traction fields is required

ρ1 v1

ρ2 v2

weldedinterface

Kinematic (displacement continuity) gives Snell’s law, but how much is reflected, how much transmitted?

http://www.walter-fendt.de/ph14e/huygenspr.htm

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Reflection & Transmission coefficients

Medium 1: ρ1,α1

Medium 2: ρ2,α2

T

Let’s take the most simple example: P-waves with normal incidence on a material interface. Dynamic conditions give:

A R

At oblique angles conversions from S-P, P-S have to be considered.

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Reflection & Transmission-Snell’s Law

What happens at a plane material discontinuity?

Medium 1: v1

Medium 2: v2

i1

i2

A special case is the free surface condition, where the surface tractions are zero.

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Free surface: P-SV-SH

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Case 1: Reflections at a free surface

A P wave is incident at the free surface ...

P PRSVR

i j

In general (also for an S incident wave) the reflected amplitudes can be described by the scattering matrix S

S =PuPd SuPdPuSd SuSd⎛ ⎝ ⎜ ⎞

⎠ 17

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Reflection and Transmission – Ansatz

How can we calculate in general the amount of energy that is transmitted or reflected at a material discontinuity?

We know that in homogeneous media the displacement can be described by the corresponding potentials

in 2-D (i.e. the wavefield does not depend on y coordinate) this gives:

and an incoming P wave has the form (aj indicate the direction cosines):

�

u = ∇Φ + ∇×Ψ

�

ux = ∂xΦ + 0− ∂zΨy

uy = 0 +∂zΨx −∂xΨz

uz = ∂zΦ +∂xΨy −0

�

Φ = A0exp i (k jxj −ωt)[ ]{ } = A0exp i ωα(ajxj − αt)

⎡ ⎣ ⎢

⎤ ⎦ ⎥

⎧ ⎨ ⎩

⎫ ⎬ ⎭

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Free surface: apparent velocity

ωp = k

x;ωη = k

z= ω 1 − sin2 i

α= ωp

cx

α

⎛

⎝⎜⎜

⎞

⎠⎟⎟

2

− 1 = ωprα

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Seismology I - Rays

R&T – Ansatz at a free surface

... here ai are the components of the vector normal to the wavefront : ai=(sin i, 0, -cos i)=(cos e, 0, -sin e), where e is the angle between surface and ray direction, so that for the free surface

where

what we know is that z=0 is a free surface, i.e.

�

Φ = A0exp ik(x−ztane− ct)[ ] + Aexp ik(x + ztane−ct)[ ]Ψ = Bexp ik' (x + ztanf−c't)[ ]

�

c = αcose

= αsini

k = ωα

cose = ωα

sini = ωc

�

c'= βcosf

k'= ωβ

cosf

�

σxz z=0 = 0σ zz z=0 = 0

P PRSVR

i je f

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Seismology I - Rays

Reflection and Transmission – Coeffs

... putting the equations for the potentials (displacements) into these equations leads to a relation between incident and reflected (transmitted) amplitudes

These are the reflection coefficients for a plane P wave incident on a free surface, and reflected P and SV waves.

�

RPP = AA0

= 4tanetanf− (1− tan2f)2

4tanetanf + (1− tan2f)2

RPSV= B

A0

= 4tane− (1− tan2f)4tanetanf + (1− tan2f)2

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Case 2: SH waves

For layered media SH waves are completely decoupled from P and SV waves

S =SuSd SuSdSuSd SuSd

⎛ ⎝ ⎜ ⎞

⎠

SH

There is no conversion: only SH waves are reflected or transmitted

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Case 2: SH waves

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Case 2: SH waves

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Case 2: SH waves

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Case 2: SH waves

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Case 2: SH waves

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Case 2: SH waves

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Case 2: SH waves

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Case 3: Solid-solid interface

To account for all possible reflections and transmissions we need 16 coefficients, described by a 4x4 scattering matrix.

Pr

SVr

PtSVt

P

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Case 4: Solid-Fluid interface

At a solid-fluid interface there is no conversion to SV in the lower medium.

Pr

SVr

Pt

P

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Ocean-Crust interface

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Crust-Mantle interface

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Body Waves and Ray Theory

Ray theory: basic principlesWavefronts, Huygens principle, Fermat’s principle, Snell’s Law

Rays in layered media Travel times in a layered Earth, continuous depth models,Travel time diagrams, shadow zones,

Travel times in a spherical EarthSeismic phases in the Earth, nomenclature, travel-time curves for teleseismic phases

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Seismology I - Rays

Basic principles

• Ray definition

Rays are defined as the normals to the wavefront and thus point in the direction of propagation.

• Rays in smoothly varying or not too complex media

Rays corresponding to P or S waves behave much as light does in materials with varying index of refraction: rays bend, focus, defocus, get diffracted, birefringence et.

• Ray theory is a high-frequency approximation

This statement is the same as saying that the medium (apart from sharp discontinuities, which can be handled) must vary smoothly compared to the wavelength.

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Seismology I - Rays

Rays in flat layered Media

Much information can be learned by analysing recorded seismic signals in terms of layered structured (e.g. crust and Moho). We need to be able to predict the arrival times of reflected and refracted signals …

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Travel Times in Layered Media

Let us calculate the arrival times for reflected and refracted waves as a function of layer depth, h, and velocities αi, i denoting the i-th layer:

We find that the travel time for the reflection is:

And for the the refraction: where ic is the critical angle:

�

sin(i1)α1

= sin(r2)α2

⇒ ic = arcsin α1

α2

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

Trefl =

2hα1cosi

= 2 h2 +X2 /4α1

�

Trefr = 2hα1cosic

+ rα2

r = X−2htanic

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Seismology I - Rays

Travel Times in Layered Media

Thus the refracted wave arrival is

where we have made use of Snell’s Law.

We can rewrite this using

to obtain

Which is very useful as we have separated the result into a vertical and horizontal term.

Trefr = Xp+2hη1

�

Trefr = 2hα1cosic

+ 1α2

X− 2hα1

α2cosic

⎛

⎝ ⎜

⎞

⎠ ⎟

�

1α2

= sinicα1

= p

�

cosic = (1− sin2ic )1 /2 = (1− p2α12)1 /2 = α1(

1α1

2 − p2)1 /2 = α1η1

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Seismology I - Rays

Travel time curves

What can we determine if we have recorded the following travel time curves?

http://www.iris.edu/hq/programs/education_and_outreach/animations/13

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Ray parameter

�

pd = sin(90°)α1

= 1α1

pr = sin(ic )α1

= 1α2

pR = sin(i)α1

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Generalization to many layers

The previous relation for the travel times easily generalizes to many layers:

Travel time curve for a finely layered Earth. The first arrival is comprised of short segments of the head wave curves for each layer.

This naturally generalizes to infinite layers i.e. to a continuous depth model.

Trefr = Xp+ 2hiηi

i=1

n

∑

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Multiple layers

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Tau-p plane

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Multiple layers

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Travel Times for Continuous Media

We now let the number of layers go to infinity and the thickness to zero. Then the summation is replaced by integration.

Now we can generalize the concept of intercept time τ of the tangent to the travel time curve and the slope p.

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Continuously varying media

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Travel Times: Examples

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Ray paths inside the Earth

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Earth’s crustAndrija

MOHOROVIČIĆ

Godišnje izvješće zagrebačkog meteorološkog opservatorija za

godinu 1909. Godina IX, dio IV. - polovina 1. Potres od 8. X. 1909

http://www.gfz.hr/sobe-en/discontinuity.htm52

Seismology I - Rays

The Inverse Problem

It seems that now we have the means to predict arrival times Tpre at a given the travel distance of a ray with a given emergence angle (ray parameter) and given structure. This is also termed a forward (or direct) problem.

We have recorded a set of travel times, Tobs, and we want to determine the structure of the Earth. Thus, what we really want is to solve the inverse problem.

In a very general sense we are looking for an Earth model that minimizes the difference between a theoretical prediction and the observed data:

where m is an Earth model.

Tobs

−Tpre(m)( )∑

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Rays in a Spherical Earth

How can we generalize these results to a spherical Earth which should allow us to invert observed travel times and find its internal velocity structure?

Snell’s Law applies in the same way:

From the figure it follows

which is a general equation along the raypath (i.e. it is constant)

sini1

v1

=sini'

1

v2

r1sini

1

v1

=r1sini'

1

v2

=r2sini

2

v2

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Seismology I - Rays

Ray Parameter in a Spherical Earth

... thus the ray parameter in a spherical Earth is defined as :

Note that the units (s/rad or s/deg) are different than the corresponding ray parameter for a flat Earth model.

The meaning of p is the same as for a flat Earth: it is the slope of the travel time curve.

The equations for the travel distance and travel time have very similar forms than for the flat Earth case!

rsiniv

= p

p = dTdΔ

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p=r0(sini)/v0

p=rP/vP

Ray Parameter in a Spherical Earth

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v0dT/(r0dΔ)=sini

p=dT/dΔ

Ray Parameter in a Spherical Earth

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Rays in homogeneous sphere

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Sphere with increasing velocity...

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Earth’s core

GNH7/GG09/GEOL4002 EARTHQUAKE SEISMOLOGY AND EARTHQUAKE HAZARD

History of Seismology 4

Richard Dixon Oldham

The Constitution of the Earth as revealed by earthquakes,

Quart. J. Geological Soc. Lond., 62, 456-475, 1906

Beno Gutenberg1914 Über Erdbebenwellen VIIA.

Nachr. Ges. Wiss. Göttingen Math. Physik. Kl, 166.

who calculated depth of the core as 2900km or 0.545R

Inge Lehmann Bureau Central Seismologique International, Series A, Travaux

Scientifiques, 14, 88, 1936.

who discovered of the earth's inner core. 60

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Earth layered structure

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Ray Paths in the Earth (1)

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Ray Paths in the Earth (2)

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Ray Paths in the Earth (3)

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Ray Paths in the Earth (4)

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Ray Paths in the Earth - Names

P P wavesS S wavessmall p depth phases (P)small s depth phases (S)c Reflection from CMBK wave inside corei Reflection from Inner core boundaryI wave through inner core

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Travel times in the real Earth

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Travel times in the Earth

Kennett, B. L. N., and E. R. Engdahl (1991). Traveltimes for global earthquake location and phase identification. Geophysical Journal International 122, 429–465.

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Spherically symmetric models

Velocity and density variations within Earth based on seismic observations. The main regions of Earth and important boundaries are labeled. This model was

developed in the early 1980's and is called PREM for Preliminary Earth Reference Model.

Model PREM giving S and P wave velocities (light and dark green lines) in the earth's interior in comparison with the younger

IASP91 model (thin grey and black lines)

A. M. Dziewonski & D. L. Anderson, 1981

http://www.iris.edu/dms/products/emc/models/refModels.htm

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